Light-Matter Response in Non-Relativistic Quantum Electrodynamics: Quantum Modifications of Maxwell's Equations

TL;DR

The paper develops a linear-response framework for non-relativistic quantum electrodynamics in the long-wavelength limit, enabling self-consistent treatment of coupled matter and quantized light within quantum-electrodynamical density-functional theory (QEDFT). By introducing Maxwell-Kohn-Sham equations and photon-exchange-correlation kernels, it reveals quantum modifications to Maxwell's equations in matter and derives a Casida-type eigenproblem for excited states that include matter-photon and photon-photon interactions. The authors validate the approach with model systems and present ab initio spectra and lifetimes for real systems such as benzene in optical cavities, capturing polariton formation, linewidths, and complex line shapes (Lorentzian to Fano) in dissipative and continuum environments. The framework thus provides a practical route to predict photon-dressed excited states, radiative lifetimes, and spectroscopic observables in polaritonic chemistry and nanophotonics, while integrating seamlessly with existing electronic-structure codes.

Abstract

We derive the full linear-response theory for non-relativistic quantum electrodynamics in the long wavelength limit, show quantum modifications of the well-known Maxwell's equation in matter and provide a practical framework to solve the resulting equations by using quantum-electrodynamical density-functional theory. We highlight how the coupling between quantized light and matter changes the usual response functions and introduces new types of cross-correlated light-matter response functions. These cross-correlation responses lead to measurable changes in Maxwell's equations due to the quantum-matter-mediated photon-photon interactions. Key features of treating the combined matter-photon response are that natural lifetimes of excitations become directly accessible from first principles, changes in the electronic structure due to strong light-matter coupling are treated fully non-perturbatively, and for the first time self-consistent solutions of the back-reaction of matter onto the photon vacuum and vice versa are accounted for. By introducing a straightforward extension of the random-phase approximation for the coupled matter-photon problem, we calculate the first ab-initio spectra for a real molecular system that is coupled to the quantized electromagnetic field. Our approach can be solved numerically very efficiently. The presented framework leads to a shift in paradigm by highlighting how electronically excited states arise as a modification of the photon field and that experimentally observed effects are always due to a complex interplay between light and matter. At the same time the findings provide a new route to analyze as well as propose experiments at the interface between quantum chemistry, nanoplasmonics and quantum optics.

Figures (9)

• Figure 1: Schematics of the Maxwell KS approach contrasted with schematics of the usual semi-classical KS theory. While in the semi-classical approach the KS orbitals are used as fixed input into the mode-resolved inhomogeneous Maxwell's equation in vacuum through the total dipole $\textbf{R}(t) = \int d \textbf{r} \, {e}\mathbf{r} \, \sum_{i} |\varphi_i(\mathbf{r},t)|^2$ (see also appendix \ref{['app:Maxwell']}), in the Maxwell KS framework the induced field acts back on the orbitals, which leads to an extra self-consistency cycle.
• Figure 2: Schematics that contrasts the usual Maxwell's equation (left) with the fully self-consistent Maxwell's equation (right). Top: The induced transversal electric field $\textbf{E}_{\perp}$ as a consequence of the induced polarization $\textbf{P}_{\perp}$, which can be equivalently expressed in terms of the auxiliary displacement field $\textbf{D}_{\perp}$. Left: mode-resolved non-self-consistent Maxwell's equation with no backreaction. The external charge current $\textbf{j}_{\alpha}$ induces the external electric field in $\textbf{E}_{\alpha}^{\textrm{tot}}=\textbf{E}_{\alpha}+\textbf{E}_{\alpha}^{\textrm{ext}}$ which acts as an external perturbation through the dipole. Since the constituents of $\tilde{\chi}^n_n$ expressed in TDDFT are purely electronic, the induced field does not couple back to the Maxwell field. Right: self-consistent Maxwell's equation in which $\textbf{j}_{\alpha}$ induces the internal field $q_{\alpha}(t)$ through the electron-photon correlated dipole which has an explicit dependence as seen in the QEDFT form of $\chi_{q_{\alpha}}^{n}$. The self-consistency of the induced field through the dipole introduces nonlinearities in the coupled system thus changes the Maxwell field at the level of linear-response.
• Figure 3: Two-level system (with excitation $\omega_0$) coupled to one mode of the radiation field (with frequency $\omega_c$). The matter subsystem is driven by an external classical field $v(t)$ and the photon mode is driven by an external classical current $j(t)$ and both subsystems are coupled with a coupling strength $\lambda$.
• Figure 4: Linear-response spectra for the extended Rabi model (dotted-red) compared to the pRPA (dashed-blue) and RWA (full-orange) approximations and for different coupling strengths $\lambda$. (a.) Absorption spectra due to matter-matter response, (b.) spectra due to photon-photon response, (c.) spectra due to matter-photon or photon-matter response. (d.) The case for $\lambda=0.7$ shows all excitations that arise in strong coupling. (a.) through (d.) describes resonant coupling. In (e.) the field is half-way detuned from atomic resonance, i.e., $\omega_{0}=2$ and $\omega_{c}=1$ with strength and energies shifted to frequencies favoring 2-photon processes. The insets in (d.) and (e.) zoom into the frequency axis showing many-photon process.
• Figure 5: Schematic of absorption spectroscopy in optical cavities: Benzene (C$_{6}$H$_6$) molecule and ${\boldsymbol\lambda_\alpha}$ denotes the polarization direction of the photon field.